The major objective of this research is to understand and visualize the structure and properties of supercoiled DNA. The trajectories of the long spatially constrained molecules are generated with curve fitting techniques commonly used in computer aided design and engineering. These approaches automatically satisfy the ring closure constraints of cyclic DNA and/or the end-to-end limitations on spatially fixed open DNA, and involve a relatively limited number of independent variables. Because of the use of mathematical representations, the number of variables that can be placed on the curves is unlimited and it is possible to treat very long chain molecules. Furthermore, because the number of independent variables is limited, it is possible to simulate the large conformational changes found in supercoiled DNAs. The immediate goal of the research is to describe chain configuration and properties in terms of realistic molecular models. The energy of the DNA will be described in terms of simple elastic terms that mimic the sequence-dependent bending and twisting of the chain observed at the local level. The proposed studies may clarify the role of local structure (primary base sequence and polyelectrolyte sugar-phosphate backbone) and ligand binding (proteins and drugs) on the overall folding of the chain. A second goal is to uncover structural details of supercoil-induced transitions of DNA, such as those involving right- and left-handed duplexes. The research combines a variety of computational approaches (Monte Carlo and molecular dynamics simulations, potential energy minimizations, finite element analysis, systematic molecular modeling) with new developments in elasticity theory. Among the scientific issues to be addressed are: (1) the competing effects of two or more bound proteins on the spatial configurations and range of flexibility of the double helix; (2) the possible role of highly bent fragments on the overall structure of kinetoplast DNA minicircles; (3) the extent to which local sequence fixes the branching in long, naturally occurring supercoils; (4) the effects of twist density and fragment length on large-scale supercoiling transitions; (5) the degree of non-uniformity of local twist in protein-bound DNAs; (6) the role of chain fluctuations on the X-ray scattering of supercoiled DNA; (7) the relative time scales of different supercoiling motions; (8) the effect of salt concentration and/or external electric field on molecular shape and flexibility.